' 



/. H. Alexander on a New Formula for Interpolations. 17 



n n + 1 n + 2 

 If we use the next expression ^ '~~7T~ ' o -? we shall de- 

 velop a 



figurate series : 1. 4. 10. 20. 35. &c. 



And the expression following would give rise to still another 



figurate series : 1. 5. 15. 35. 70. &c. 



It is seen in the algebraic notation of these, that not only the 

 individual terms of each, but the general terms of all, are in a 

 regular progression. Thus the first or natural series, compre- 

 hends only the lowest power; the next involves the square of 



the given term and so presents an equation of the second degree ; 



the third offers an equation of the third degree : and so on. This 

 is precisely accordant with what was just now deduced in the 

 consideration of straight lines and curves. 



The powers of n, occurring in these different equations, corres- 

 pond also to the degrees of removal from the original series at 

 which the differential series becomes constant. Thus taking again 

 the natural series : 1. 2. 3. 4. 5. &c. 



1st differential do. : 1. 1. 1. 1. &c. 



That is to say, the differences are already constant at the first 



remove. 



If we take a series of the second order, involving the second 



power of n, we have 



second order: 1. 3. 6. 10. 15. &c. 



1st differential : 2. 3. 4. 5. &c. 



2d differential : 1. L L &c. 



and the differences become constant at the second remove. 



These instances are sufficient to show the generality of what 

 is affirmed ; that the index of the power involved in the equation, 

 indicates also the number of removes at which the differences of 

 the terms become constant. It is upon this, that the ordinary 

 method of interpolation is founded j but the distinction between 

 that method and the formula now proposed, is that while the 

 former requires the use of the differences themselves and in cases 

 of intermediate interpolation the tabulation of the results, the 

 latter employs in all cases only the original terms, and uses the 

 differences only so far as may serve in the absence of other 

 means to determine the degree of the equation and of the figu- 

 rate series in which the quantities given are to be arranged. 



The order of the series and the number of terms necessary for 

 solution are, it is manifest, correlatives ; and the law which con- 

 nects them is, that the index of the order -f 1 is the said num- 

 ber. Thus, in a series of the first order, the given quantities in 

 which, progress lineally or by one constant difference, only two 

 terms are necessary for interpolation of the remaining terms; 



Second Series, Vol. VII, No, 10, Jan., 1849. 3 



